Distinct roots

Number of distinct root loci This is equal to the order of the eharaeteristie equation. 

For C > 1 (overdamped system). If the damping coefficient is greater than unity, the quantity inside the square root is positive. Then St and S2 will both be real numbers, and they will be different (called distinct roots). 

These are the only two possibilities. We cannot have three complex roots. The complementary solution would be either (for distinct roots)... 

In the general case, if A is not a normal matrix, then it is not necessarily diagonalizable. However, it is diagonalizable if the characteristic equation has n distinct roots. 

Equation (2.29) has one distinct root, x2=0, and hence the above singularly perturbed ODE system is in standard form. Proceeding with our analysis, we obtain a reduced-order, uniform approximation of the slow component of the dynamics as... 

There are two distinct roots for rj in Equation 7 which represent two positions on the ellipse at which the maximum and minimum values of... 

Typically when linear partial differential equations are solved using the Laplace transform method the solution obtained in the Laplace domain can be represented as in equation (8.7) and q(s) usually has an infinite number of roots. If s = p . n = 1..00 are the distinct roots of q(s), q(s) can be factorized as... 

If. ., r are distinct roots of the characteristic equation, then the general solution is given by... 

The polynomial P(s) is of second order and has two distinct roots which are not real (as in the previous case) but complex conjugates ... 

Compute C4. This constant corresponds to the distinct root and can be computed using the procedure described earlier. Thus multiply both sides of (8.19) by (s + 2) and then set 5 + 2 = 0 (i.e., j = -2) and find that C4 = -1. 

The complementary solution would be either (for distinct roots)... 

A second major assumption in the phylogenetic map is that there must be a distinct root race and an Eve, a single point start to humanity. This assumption is incorrect. First, consider the age-old paradox of what came first, the hen or the egg. Of course, the answer is an earlier kind of hen. ° Those of us that secretly harbour a romantic conception of Eve might need to catch their breath because around 65 to 125 million years ago, Eve was a rat-like creature called aMus musculus. This Eve is not only the ancestor of humans, but also the common ancestor of humans and mice. ... 

The Hiickel matrix of order 2N is symmetrical and tridiagonal, and therefore has exactly 2N non-degenerate real eigenvalues. Since equation (25) is, a part from a constant factor, the secular determinant of the Hiickel matrix [6], it has exactly N non-zero distinct roots. 

EX4iMPL 3.1 DISTINCT ROOTS (NOT DIFFERING BY INTEGER, CASE I) Consider the second order equation with variable coefficients... 

To adequately display the features of the plot, the CSTR locus is only shown up to a maximum residence time of 30 h (the trend observed at 30 h remains unchanged for longer residence times). A distinct root is seen on the graph at T 1.33 h, signifying a critical CSTR point. However, for CSTR residence times greater than approximately 10 h, the value of A(C) lies very close to zero (and only truly reaches... 

For an nth-order equation, which has m repeated roots (A,) and one distinct root (A ), the... 

A single root of the interpolator follows the surface, but the other roots may cross the interpolating root or run close to it. In many applications this is not a difficulty since it is quite easy to pick out the interpolating root by inspection and follow it. So far, we have not encountered a case in which two distinct roots provide representations or two different regions of the same branch of the same surface. In applications which require the interpolator to... 

These and other oidiogonal polynomials have many properties in common. We shall not attempt to explore in detml their common properties here, but we note (without proof) the nontrivial fact that an orthogonal polynomial p x) of degree n has n distinct roots or zeros... 

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